/*
*
给定一个包含非负整数的 m x n 网格 grid ，请找出一条从左上角到右下角的路径，使得路径上的数字总和为最小。

说明：每次只能向下或者向右移动一步。

示例 1：

输入：grid = [[1,3,1],[1,5,1],[4,2,1]]
输出：7
解释：因为路径 1→3→1→1→1 的总和最小。
示例 2：

输入：grid = [[1,2,3],[4,5,6]]
输出：12

提示：

m == grid.length
n == grid[i].length
1 <= m, n <= 200
0 <= grid[i][j] <= 200

  - @author ala
  - @date 2024-09-18 15:27
*/
package main

import (
	"fmt"
	"math"
)

func main() {
	grid := [][]int{{1, 3, 1}, {1, 5, 1}, {4, 2, 1}}

	fmt.Println(minPathSum(grid))
}

func minPathSum(grid [][]int) int {
	return V1(grid)
}

var INF int = math.MaxInt >> 1

/**
 *	1）dp[i][j] = min(dp[i - 1][j] + grid[i - 1][j - 1], dp[i][j - 1] + grid[i - 1][j - 1])
 */
func V1(grid [][]int) int {
	M, N := len(grid), len(grid[0])
	dp := make([][]int, M+1)
	for i := range dp {
		dp[i] = make([]int, N+1)
		dp[i][0] = INF
	}
	for i := range dp[0] {
		dp[0][i] = INF
	}
	dp[0][1] = 0
	dp[1][0] = 0

	for i := 1; i <= M; i++ {
		for j := 1; j <= N; j++ {
			dp[i][j] = min(dp[i-1][j]+grid[i-1][j-1], dp[i][j-1]+grid[i-1][j-1])
		}
	}
	return dp[M][N]
}
